30-60-90 and 45-45-90 — memorize once, solve instantly
Reviewed by [email protected], Geometry Calculator Developer & Online Math Educator Last updated May 21, 2026
Two right triangles appear so often in geometry, trigonometry, and physics that the side ratios are worth memorizing: the 30-60-90 triangle (sides 1 : √3 : 2) and the 45-45-90 triangle (sides 1 : 1 : √2). Once you recognize the angles, you can read off the sides without computing — saving a Pythagoras calculation every time.
| Name | Formula | Notes |
|---|---|---|
| 30-60-90 Side Ratio | 1 : √3 : 2 |
Opposite the 30°, 60°, 90° angles respectively. Short leg, long leg, hypotenuse. |
| 30-60-90 Long Leg | long_leg = short_leg × √3 |
The side opposite 60° is √3 times the side opposite 30°. |
| 30-60-90 Hypotenuse | hypotenuse = 2 × short_leg |
The hypotenuse is always exactly twice the shortest side (opposite 30°). |
| 30-60-90 Short Leg from Hyp | short_leg = hypotenuse / 2 |
Inverse: if you know the hypotenuse, half of it gives the side opposite 30°. |
| 30-60-90 Long Leg from Hyp | long_leg = (hypotenuse / 2) × √3 |
Substitute short_leg = hyp/2 into long_leg = short × √3. |
| 45-45-90 Side Ratio | 1 : 1 : √2 |
Both legs are equal (isosceles right triangle); the hypotenuse is leg × √2. |
| 45-45-90 Hypotenuse | hypotenuse = leg × √2 |
For any equal-legs right triangle, the hypotenuse is √2 ≈ 1.414 times either leg. |
| 45-45-90 Leg from Hyp | leg = hypotenuse / √2 = hypotenuse·√2/2 |
Inverse — when you know the hypotenuse, divide by √2. |
| 45-45-90 from Square | half a square diagonally = 45-45-90 |
A 45-45-90 triangle is exactly half of a square cut along its diagonal. This is why s² + s² = (s√2)² = 2s². |
| 30-60-90 from Equilateral | half an equilateral cut by altitude |
A 30-60-90 triangle is half of an equilateral triangle cut by its altitude. This is where the ratios 1 : √3 : 2 come from. |
| sin(30°), cos(60°) | = 1/2 |
Read directly from the 30-60-90 triangle: opposite/hypotenuse = short/2 = (1)/(2) when short = 1. |
| sin(60°), cos(30°) | = √3/2 |
Opposite/hypotenuse = long/2 = √3/2 in the 1:√3:2 triangle. |
| sin(45°), cos(45°) | = √2/2 ≈ 0.707 |
In a 45-45-90, the angle's opposite and adjacent are both legs (both = 1), and hypotenuse = √2. So sin = 1/√2 = √2/2. |
| tan(30°) | = 1/√3 = √3/3 |
opposite/adjacent = short/long = 1/√3. |
| tan(45°) | = 1 |
opposite/adjacent = leg/leg = 1. |
| tan(60°) | = √3 |
opposite/adjacent = long/short = √3/1. |
Efficiently solving for unknown dimensions in special right triangles requires understanding the specific input parameters for each triangle type. Our calculator simplifies this process by allowing you to derive all missing sides and angles from a single known value. To begin, identify which type of triangle you are working with: either a 45-45-90 triangle or a 30-60-90 triangle. These shapes have fixed angle measures and consistent side ratios, which allows for precise calculations without needing complex trigonometric functions.
First, enter the known side length into the designated input field. This value can be either a leg or the hypotenuse, depending on the information provided in your problem. For a 45-45-90 triangle, the two legs are always equal in length. If you know the length of one leg, the other leg is identical. If you know the hypotenuse, you can find the leg length by dividing the hypotenuse by √2. Conversely, for a 30-60-90 triangle, the sides follow a ratio of 1 : √3 : 2. The shortest side is opposite the 30-degree angle, the longer leg is opposite the 60-degree angle, and the hypotenuse is twice the length of the shortest side.
After entering your value, select the corresponding triangle type from the dropdown menu. This ensures the calculator applies the correct geometric relationships. Finally, click the calculate button. The tool will instantly compute the remaining side lengths and confirm the angle measurements. While these calculators rely on fixed ratios, they are mathematically equivalent to using trigonometric ratios like sine, cosine, and tangent, but with significantly fewer steps. Additionally, you can verify your results manually using the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (a² + b² = c²). This step-by-step approach ensures accuracy and saves time when solving geometry problems involving these specific triangular forms.
Even with standardized shapes, errors frequently occur when applying geometric principles to 30-60-90 triangle and 45-45-90 triangle problems. One of the most prevalent issues involves confusing the roles of the legs and the hypotenuse. In a 45-45-90 triangle, students often mistakenly assume the hypotenuse is equal to the leg length multiplied by two. However, the correct relationship is that the hypotenuse is the leg length multiplied by √2. Similarly, in a 30-60-90 triangle, the hypotenuse is twice the length of the shorter leg, not the longer leg. Misidentifying which side corresponds to which part of the ratio leads to significant calculation errors.
Another common pitfall is misapplying side ratios across different triangle types. The ratio 1 : √3 : 2 is exclusive to the 30-60-90 configuration, while the 45-45-90 triangle follows the 1 : 1 : √2 pattern. Using the wrong ratio set invalidates the entire solution. It is crucial to recognize the angle measures first before selecting the appropriate mathematical model. Furthermore, many learners forget that the hypotenuse is always the longest side in any right triangle. This fundamental property serves as a quick check for validity; if a calculated hypotenuse is shorter than a leg, the result is physically impossible. Relying solely on memorized shortcuts without verifying against the Pythagorean theorem can also lead to mistakes. Understanding the underlying geometry rather than just the formula helps prevent these errors. By carefully distinguishing between the leg and the hypotenuse and respecting the unique properties of each triangle type, you can ensure accurate results in your geometric computations.
Plug in your numbers and get instant step-by-step results.